C 505 ] 
take A N equal to B H, and draw N O paral- 
lel to AC, BD, and H I. 
Then if N P be drawn, that P O be to ON in 
the given ratio, N P will be given in pofition, and 
P O will be to O N, that is, E G, as the rectangle 
under KEL to that under EM, E G ; fo that the 
rectangle under KEL will be equal to that under 
PO, EM. But the redangle under O K M is equal 
to the excefs of that under OEM above that under 
KEL [d 1 ; therefore the redangle under O K M, 
or that under N A H, or under NBH, is equal to 
that under E M and the excefs of OE above O P, 
that is, to the redangle under PEM; the point E 
therefore is in an hyperbola defcribed to the given 
alymptotes PN, MH, and paffing through A and B. 
Again if two of the given lines only are parallel, 
but the redangles otherwife related to them, than 
as above. Suppofe the ratio of the redangle under 
A G, E F to that under B G, GE is given. Let 
C D meet A B in L, and let H E I, M F N be 
drawn parallel to A B, and L K parallel to A C and 
B D. Then the parallelogram E M will 
be to the parallelogram EB in the given I0 * 
ratio. Take AO to OB in that ratio, and draw 
OP parallel to A C and BD. Here the point O 
will be given, and the parallelogram P A will be in 
the given ratio to the parallelogram P B ; whence 
A B will be to B O as the parallelogram B H to the 
parallelogram B P, and as the difference between 
the parallelograms E M and E B to the parallelo- 
gram E B,confequently as the parallelogram G M to 
the parallelogram P G ; therefore the ratio of the 
redangle under A G, F G to the redangle under 
[d~] By Prop. 194. Lib. 7. Papp. 
